Squares and Square Roots: An Introduction
After you are done with the preceding chapters, you will proceed to learning about squares and square roots in your CBSE Maths Class 8 Chapter 6 curriculum. This is one of the most interesting chapters that requires you to concentrate properly. Along with the various mathematical solutions, this chapter also helps a student understand the various properties of square roots. Additionally, you can learn various tips and tricks regarding the calculation of perfect squares and square roots.What are Perfect Squares?When you multiply any whole number with itself, you get a result, which is known as the perfect square of the first number.Let us consider an example to clarify what a perfect square signifies. Suppose, the starting number is 12. When multiplied by itself or squared, the resulting figure is 144. You can say that 144 is the perfect square for the number 12.Important Properties of all Square NumbersThe key to understanding squares and square roots is to first gauge a few common characteristics, which are shared by all square numbers. Learning them can help your calculations greatly – If a number has 3, 8, 2 or 7 in its units’ place, then it cannot be a perfect square number.Numbers with 0, 4, 1, 6, 5 and 9 may or may not be perfect squares. For example, 36 has six in the units’ place and is the square of the number 6. Nevertheless, 136 also has six in the units’ place but is not a perfect square. Its square root is 11.66.Square Root CalculationWhen you know the square of a number but do not know the number itself, the procedure to discover this number is known as square root. It is basically the inverse function of squaring a figure.Thus, now that you know the relationship between square and square roots, here is a few tips that should assist you in estimating square root from a given figure.Useful NotesYou can easily calculate how many digits the square root of a given perfect square possesses by a simple formula.Consider ‘x’ as the number of digits in the perfect square. Number of digits (n) in the square root is equal to x/2, where x is even. If x is odd, n = x+1x+1x + 1/2.For example, let us consider the number 625. Here, x = 3, which is an odd number. Therefore, n = 3+13+13 + 1/2 = 2. We can confirm this assertion as the square root of 625 is 25, which has two digits.Estimating the Value of Square RootsIf you are in a hurry and need an approximate square root value, you can estimate the same using a simple technique. To demonstrate the same, let us take the number 671 as an example. It is not a perfect square. However, to estimate, let us consider the two perfect square numbers closest to 671, which are 625 and 676. Since 671 is closer to 676, whose square root is 26, you can assume that the square root of 676 is approximately 26 as well. Calculating Square Root Using Subtraction MethodIn squares and square roots chapter, you will also learn about another method to derive square root from a perfect square. Suppose, the number is 25. The number of subsequent odd number subtractions you can undertake from the given number before reaching 0 should be its square root.25 – 1 = 2424 – 3 = 2121 – 5 = 1616 – 7 = 99 – 9 = 0As you can see, there are 5 such subtractions before reaching 0. Therefore, we can say that the square root of 25 is 5. Multiple Choice QuestionQ. Which of the following numbers will produce even squares?981783268547Ans. (c) 268Are you facing difficulty understanding any section of the squares and square roots chapter? Well, we at Vedantu can help you with all your preparations. Our online classes and excellent study materials are designed to give students an edge. Moreover, with doubt clearing sessions, our students can attain extensive clarity on each topic. Come and join Vedantu for a class on perfect squares and square roots!
Need of Square and Square Roots
Not only in everyday math but also in advanced stages of life square and square roots are applicable. square roots are not only important topics for students but this is applied by engineers at their work. square and square roots are applicable for getting quadratic equations and solving Pythagoras theorem. Pythagoras Theorem is based on square roots of a number. There is a need for square and square roots because it is not only applicable to math but it is very useful for deriving standard deviation in statistics; it is useful in Algebra as well as in geometry.
Square and square roots are really very important for math students not only at school level but also during preparation of Neet for exams. square and square roots are introduced to the students from class 7 onwards but students learn more about them in class 8. CBSE books of Standard 8 explains this particular topic in detail:
The contents of the course book are:
Rational numbers
Linear equations in one variable
Understanding quadrilaterals
Practical geometry
Data handling
Square and square roots
Cubes and cube roots
Comparing quantities
Algebraic expressions and identities
Visualizing solid shapes
Area of polygons surface
Area and volume of solids
Exponents and powers
Direct and Inverse proportions
Factorization introduction to graphs
Playing with numbers.
The Chapter 6 Explains Regarding:
squares of a number, square of a rational number, square of a large number, finding the product of two consecutive odd-even integers, finding a Pythagorean triplet, it also explains regarding finding square roots by various methods, finding square root through repeated subtraction, finding square root through prime factorization, finding square root by division method, properties of square root and approximate square root by division method.
FAQs on Squares and Square Roots
1.What are the properties of a square number?
The following are the properties of a square number:
The number which ends in 2,3,7 or 8 is never a perfect square.
the unit digit of the square of a number can be determined if the unit digit of a number is known.
the number of zeros at the end of a perfect square is always even. the square of an even number is always even .
the square of an odd number is always or the sum of the first N odd natural numbers is equal to n square.
the square of any or natural numbers other than one can be expressed as the sum of two consecutive positive integers.
if 1 is added to the product of two consecutive odd natural numbers the result obtained is equal to the square of only even natural numbers between them.
if 1 is added to the product of two consecutive even natural numbers then the result obtained is equal to the square of the only odd natural numbers between them.
three natural numbers m n and P are said to form a Pythagorean triplet.
2.How can we find the square root of a number through prime factorization?
when a given number is a perfect square we can find the square root through Prime factorisation using the following steps.
- step 1: find all prime factors of the given number
- step 2: group the similar factors into pairs
- step 3: select one prime factor from each group and multiplied them their product is a square root of the given number.
3.Which method of square root is useful for large numbers?
For finding square root for large numbers division method is more useful. Prime factorization method becomes very cumbersome and time consuming to evaluate square roots for a larger number. so the division method is really very helpful for large numbers. The long division method can be extended to any number .However large it is and even two numbers given in decimal forms the number of digits in square root is always equal to the number of groups in the given square number. Hence the long division method is more useful for deriving square roots rather than other types of methods.
4.How square and square root methods are useful for economists?
Square and square roots are not only helpful for mathematicians for solving algebraic equations and Pythagoras Theorem but they are very useful for others like engineers including the Economist also. Economists Use them while calculating the frequency of distribution through standard deviation.It is used in standard deviation as it is based on square and square root principles. pertaining to the results in statistics, a standard deviation method is used. The entire standard deviation chapter depends on the calculation of data based on the principles of square and square roots hence square and square roots are helpful for even economists.
5. From where can students get more important questions regarding square and square roots?
Students can get solutions from CBSE textbooks itself where the entire chapter is thoroughly explained. students can also go through NCERT solutions of 8std Math provided at the Vedantu website. Therefore, it will be easy for students to revise the chapter with this material and score in the final examination.